 Radial Schwarzschild escape with asymptotic speed

For asymptotic speed $0\le u<c$, $k=(1-u^2/c^2{)}^{-1/2}$ and the local outward speed obeys $v_{\mathrm{loc}}^2=u^2+(2GM/r)(1-u^2/c^2)$. It tends to $c$ as the horizon is approached from outside, though a static observer cannot actually remain on the horizon.